, Volume 50, Issue 1, pp 4661
First online:
Growth in Poisson algebras
 S. M. RatseevAffiliated withUl’yanovsk State University Email author
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Get AccessA criterion for polynomial growth of varieties of Poisson algebras is stated in terms of Young diagrams for fields of characteristic zero. We construct a variety of Poisson algebras with almost polynomial growth. It is proved that for the case of a ground field of arbitrary characteristic other than two, there are no varieties of Poisson algebras whose growth would be intermediate between polynomial and exponential. Let V be a variety of Poisson algebras over an arbitrary field whose ideal of identities contains identities {{x _{1}, y _{1}}, {x _{2}, y _{2}}, . . . , {x _{ m }, y _{ m }}} = 0 and {x _{1}, y _{1}} · {x _{2}, y _{2}} · . . . · {x _{ m }, y _{ m }} = 0, for some m. It is shown that the exponent of V exists and is an integer. For the case of a ground field of characteristic zero, we give growth estimates for multilinear spaces of a special form in varieties of Poisson algebras. Also equivalent conditions are specified for such spaces to have polynomial growth.
Keywords
Poisson algebra growth of variety colength of variety Title
 Growth in Poisson algebras
 Journal

Algebra and Logic
Volume 50, Issue 1 , pp 4661
 Cover Date
 201103
 DOI
 10.1007/s104690119123z
 Print ISSN
 00025232
 Online ISSN
 15738302
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Poisson algebra
 growth of variety
 colength of variety
 Authors

 S. M. Ratseev ^{(1)}
 Author Affiliations

 1. Ul’yanovsk State University, ul. L. Tolstogo 42, Ul’yanovsk, 432970, Russia